Effect of Turbulence on the Disintegration Rate of Flushable Consumer Products Fatih Karadagli1*, Bruce E. Rittmann2, Drew C. McAvoy3, John E. Richardson4
ABSTRACT: A previously developed model for the physical disintegration of flushable consumer products is expanded by investigating the effects of turbulence on the rate of physical disintegration. Disintegration experiments were conducted with cardboard tampon applicators at 100, 150, and 200 rotations per minute, corresponding to Reynold’s numbers of 25 900, 39 400, and 52 900, respectively, which were estimated by using computational fluid dynamics modeling. The experiments were simulated with the disintegration model to obtain best-fit values of the kinetic and distribution parameters. Computed rate coefficients (ki) for all solid sizes (i.e., greater than 8, 4 to 8, 2 to 4, and 1 to 2 mm) increased strongly with Reynold’s number or rotational speed. Thus, turbulence strongly affected the disintegration rate of flushable products, and the relationship of the ki values to Reynold’s number can be included in mathematical representations of physical disintegration. Water Environ. Res., 84, 424 (2012).
shear rate and, hence, the rate of disintegration of a solid product. Following disintegration, hydrolysis of solids occurs because of attacks of nucleophiles, such as OH2 or H2O, that break ester bonds between organic molecules and result in release of organic alcohols and acids (Stumm and Morgan, 1996). Hydrolysis may be aided by extracellular enzymes. Biodegradation transforms the soluble molecules to simple end products like carbon dioxide and water (Rittmann and McCarty, 2001). In previous work by Karadagli et al. (2009), the physical breakup of solids was represented with a disintegration rate coefficient (ki) whose value depends on the mechanical strength of products and with a Reynold’s number to represent turbulence, as follows:
KEYWORDS: flushable consumer products, turbulence, disintegration constants, simulation.
Rdis ~
dM dC ~V ~ki |Re|C|V dt dt
ð1Þ
doi:10.2175/106143012X13354606450960
Where
Introduction A large number of flushable consumer products, such as toilet tissue and tampons, are marketed around the world. Flushable consumer products often are made out of cotton and/or cellulosic fibers that are bonded together with water-soluble polymers. These products are typically disposed of in sewer systems, where they undergo various degradation processes such as physical disintegration caused by turbulent forces, hydrolysis of soluble material, and biodegradation of dissolved materials. Physical disintegration depends on turbulence and solid characteristics such as size, mechanical strength, and density of solids. Smaller and low-density particles flow with water streamlines and avoid shear stresses caused by turbulence, while large and dense solids are more subject to shear as they cross streamlines. Mechanical strength determines how fast the product disintegrates. Turbulence in water, which can be represented with the Reynold’s number, should increase the
1 * Department of Environmental Engineering, School of Engineering, Sakarya University, Esentepe, Sakarya, Turkey; telephone: 90-2642955636; facsimile: 90-264-2955601; e-mail:
[email protected]. 2
Swette Center for Environmental Biotechnology, Biodesign Institute at Arizona State University, Tempe, Arizona.
3
School of Energy, Environmental, Biological, and Medical Engineering, University of Cincinnati, Cincinnati, Ohio.
4
Blue Hill Hydraulics Inc., Blue Hill, Maine.
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Rdis 5 the rate of change in mass of the solid product due to physical disintegration (g/s), M 5 mass of the product (g), C 5 concentration of the solid product (g/L), V 5 liquid volume containing the solid product (L), ki 5 disintegration rate coefficient (h21) that depends on the solid’s mechanical properties, and Re 5 Reynold’s number representing turbulence in water (unitless), which is defined by Re~ðr|v|dÞ=m
ð2Þ
Where r 5 water density (g/L), v 5 average flow velocity (cm/s), d 5 depth of water (cm), and m 5 viscosity (g/cm?s). When the liquid volume is constant over time, the total mass (M) of a solid in a system equals the product C 3 V, and the total mass in the system can be represented in terms of a product’s mass concentration. Karadagli et al. (2009) tested the ability of the disintegration model shown in eq 1 to simulate the fate of the following two different flushable products in batch disintegration tests: Products A and B, which had water soluble contents of 40% and 0%, respectively. The model was able to capture disintegration of the original large solid and the sequential production of small-sized ‘‘chips’’. From the experimental data, Karadagli et al. (2009) estimated the size-distribution fractions for particles Water Environment Research, Volume 84, Number 5
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disintegrating from a large solid and the disintegration rate coefficients for each size fraction. The disintegration model captured and quantified the distinctly different disintegration patterns for the two flushable solid products. Here, this study investigates the effect of turbulence on physical disintegration with batch-disintegration experiments conducted with a cardboard tampon applicator at three different turbulence conditions. Using the experimental results and an estimate of turbulence from computational fluid dynamics (CFD) simulations of the test system, disintegration rate coefficients (ki) for each solid size range and distribution ratios of solids were estimated through modeling analyses. This study also presents the relationship between rate coefficients and turbulence, represented by the estimated Reynold’s number. This assessment shows that the disintegration rate increases as turbulent forces increase in the system and that the disintegration model can simulate the outcome of the experiments for the range of turbulence conditions. Materials and Methods Experimental Procedures. Baffled 2.8-L Fernbach flasks (Bellco Glass, Vineland, New Jersey) were used to carry out batch disintegration experiments. A Fernbach flask has a diameter of 15.3 cm at the bottom and gently rounds up and out by 2.5 cm, reaching a maximum diameter of 20.3 cm at a height of 4.1 cm from the bottom. From here, the flask slopes inward to a final diameter of 6.3 cm (16.5 cm from the bottom). The flask then goes straight up an additional 6.3 cm to the top. Thus, the total height of the flask is 22.9 cm. In addition, three evenly spaced 1.25-cm baffles were cut into the bottom of the flask where it curves from the 15.3-cm diameter to a diameter of 20.3 cm. In the experiments, raw wastewater was used to simulate the conditions that would occur during sewer conveyance. The raw wastewater was collected from the influent of a municipal wastewater treatment plant and screened through a 1-mm sieve prior to use. One liter of prescreened raw wastewater was placed into the flask and 1.5 to 3 g/L of test material (i.e., cardboard tampon applicator) was added. The experiment was conducted with a cardboard tampon applicator in the dark at ambient room temperatures. The experimental system of this study consisted of flasks mounted on an orbital shaker table (LAB-LINE Instruments, Melrose Park, Illinois) that had a 2.54-cm orbit. The flasks with the test material were subjected to rotation speeds of 100, 150, or 200 rpm. One of the flasks was removed for sampling at the following time points: 0.25, 0.5, 1, 2, 4, 6, 24, and 48 hours. During sample analyses, the entire content of each flask was passed through a series of USA Standard Testing Sieves (VWR Scientific, West Chester, Pennsylvania) with an 8-, 4-, 2-, and 1-mm opening. Each sieve was rinsed to collect the material retained, which was transferred to an aluminum weigh pan, dried overnight at 40 uC, desiccated, and weighed on a four-place analytical balance (Sartorius AG, Goettingen, Germany). The material on each sieve is represented as .8 mm (i.e., solids on 8mm sieve); 4 to 8 mm (i.e., solids on 4-mm sieve); 2 to 4 mm (i.e., solids on 2-mm sieve); 1 to 2 mm (i.e., solids on 1-mm sieve); and ,1 mm (i.e., solids smaller than 1 mm). The experiments at 150 rpm were carried out in triplicates to estimate experimental reproducibility. Experiments at 100 and 200 rpm were carried out once. May 2012
Computational Fluid Dynamics Modeling. Flow velocities in a shake flask vary constantly during rotational movement. Thus, flow velocities in a shake flask could not be measured with experimental methods. Similarly, flow velocities cannot be estimated by using analytical equations for flow movement. Therefore, three-dimensional simulation of fluid motion in the experimental setup is a feasible method to estimate hydraulic parameters in the experimental system. Accordingly, CFD modeling techniques are used to simulate and analyze flow patterns within the shake flask. The results of this CFD work indicated turbulence levels in water at different rotational speeds, which are used to evaluate the effect of turbulence on the disintegration rate of flushable consumer products. Computational fluid dynamics modeling techniques for the simulation transient free-surface flows are commonly used to solve a wide variety of complex flow phenomenon (Anderson, 1995). This modeling approach was suitable for this study because it allowed for evaluation of flow characteristics between different scenarios in the experimental system. For example, turbulence in the flask increases along with rotational speeds and the model results capture these changes well (e.g., the calculated movement of water within the flask seems to match the observed movement of water within the flask). Based on results such as these, it was plausible to draw conclusions about the likely effect of turbulence on disintegration rates. It should be noted that CFD programs do require estimation of certain parameter values, particularly those related to turbulence closure; however, in this study, standard accepted values were used that remained the same for all our simulations. Subsequently, CFD analyses were based on a comparison of simulation results where all setup parameters remained the same except for one, that is, the rotational rate of the flask. Therefore, a side-by-side comparison of the study’s simulations provides a feasible way to investigate effects of turbulence on disintegration of flushable products. In this study, the CFD software, FLOW-3DH Version 7.2 (Flow Science Inc., Santa Fe, New Mexico), was used to simulate water motion in the shake flask at different rotational speeds and to estimate the parameters needed to define the Reynold’s number (Sicilian et al., 1987). The flask was divided into computational elements in which transient velocities were calculated from three-dimensional Navier–Stokes equations of fluid motion. The fractional area/volume obstacle representation method was applied to define obstacles within the computational mesh (Hirt and Sicilian, 1985) and the volume of fluid method was applied to define the initial fluid configuration inside the flask (Hirt and Nichols, 1981). Flow in the numerical model was controlled by boundary conditions set at the fluid free surface (i.e., pressure 5 1 atm and zero shear). The renormalization-group method was used for turbulence closure (Yakhot and Orszag, 1986; Yakhot and Smith, 1992). In addition, no-slip conditions were specified at solid boundaries. In all instances, the computational mesh was filled with 1 L of water and all flow velocities were initially set to zero. For shake flask simulations, a three-dimensional computeraided drawing (CAD) of the test flask was created for use directly in the model setup. The CAD drawing was constructed with the software, AUTOCADH (Autodesk Inc., San Rafael, California), using physical dimensions of the flask previously described. 425
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Table 1—Set of equations used in FLOW3DH to compute selected key parameters. Parameter*
Formula P
Specific kinetic energy
mi |vi 2 M V ~SQRT (KE ) V |L|r Re~ m KE ~
Characteristic flow velocity Reynold’s number
Units (Length/time)2 Length/time Unitless
* Parameter definitions and units: mi 5 mass of water in the ith cell in units of mass (g); vi 5 velocity in the ith cell in units of length/time (cm/ s); M 5 total mass of water in the mesh, (g); KEmean 5 estimated mean kinetic energy (cm2/s2); V 5 average velocity (cm/s); L 5 diameter of the flask (cm); r 5 density of water (g/cm3); and m 5 dynamic viscosity of water (g/cm?s).
To simulate the effects of the shaker table, the direction of the gravitational constant was continuously changed during the CFD simulation. This adjustment was made by using the non-inertial reference-frame algorithms resident in the standard FLOW-3DH package. This approach allowed us to mimic the acceleration of the wastewater in the beaker. In this analysis, a quasi-steadystate was achieved at 0.7 seconds of a 2.4-second time period, and steady-state values were used to perform computations of specific kinetic energy. Key hydraulic parameters and the mathematical expression used for estimating the Reynold’s number are presented in Table 1. Initially, kinetic energy of a cell was estimated in the simulation mesh from (mivi2). Then, these values were summed and the sum was divided with the total mass of fluid in the flask to determine the specific kinetic energy of the system; this is presented in the first equation in Table 1. This value is not the typical kinetic energy 5 (K)*m*v2; instead, it represents the average ‘‘specific kinetic energy’’ of the experimental system. Accordingly, units of this equation are not in typical kinetic energy units, (mass*(length/time)2), but rather in ‘‘specific kinetic energy’’ units, or (length/time)2. Specific kinetic energy values of the system from eq 1 were estimated at every time interval over the simulation period of 0 to 2.4 seconds. Figure 1 illustrates how the specific kinetic energy of the system changed with time. From the oscillations of specific kinetic energy at steady state, minimum, maximum, average [(minimum + maximum)/2], and mean specific kinetic energy values of the
Figure 1—Estimated specific kinetic energy (cm2/s2) of the system during simulation period(s). The pattern reaches quasi steady state at approximately 0.7 seconds. system were picked for the simulation period at a given rotational speed. The mean specific kinetic energy value is the statistical mean value that is estimated by dividing the sum of specific kinetic energy values with the number of data points. Then, the characteristic flow velocity in the flask was estimated by using the second equation in Table 1, that is, V 5 SQRT(KE). Lastly, the Reynold’s number was calculated from the characteristic flow velocity using the third equation in Table 1. Mathematical Modeling of Disintegration. Details of the mathematical model for disintegration used in this study were previously presented by Karadagli et al. (2009); thus, the basic structure of the mathematical model is briefly presented. In the conceptual model of disintegration, a flushable solid product disintegrates and produces smaller-size chips in various sizes and fractions. The main product (e.g., .8-mm solid) generates 4- to 8-mm, 2- to 4-mm, 1- to 2-mm, and ,1-mm size solids in various amounts. Similarly, the intermediate-size solids (i.e., 4 to 8 and 2 to 4 mm) produce solids of 1 to 2 mm and ,1 mm. The form of eq 1 was used to develop equations for chip sizes of . 8-mm, 4- to 8-mm, 2- to 4-mm, 1- to 2-mm, and ,1-mm solids (see eqs 3 through 7). Fractional distribution ratios (fj) were introduced to the rate equations to indicate how much of the
Table 2—Specific kinetic energies (cm2/s2) vs rotations-per-minute values obtained with CFD modeling over a 2.4-second simulation.
Rotations per minute (1/min) 64 75 100 150 200 250 300 1 2 3
426
Minimum value of estimated specific kinetic energy (cm2/s2)
Maximum value of estimated specific kinetic energy (cm2/s2)
Range of specific kinetic energy1 (cm2/s2)
Average specific kinetic energy2 (cm2/s2)
Mean specific kinetic energy3 (cm2/s2)
13 60 397 614 319 218 162
31 177 680 931 914 1549 1566
18 117 283 317 595 1331 1404
22 119 539 773 617 884 864
24 114 538 765 568 831 805
Range of specific kinetic energy is estimated from maximum value minus minimum value. Average specific kinetic energy is estimated as (maximum value + minimum value)/2. Mean specific kinetic energy is estimated by dividing the sum of specific kinetic energy values with the total number of data points. Water Environment Research, Volume 84, Number 5
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Figure 2—Fluid Movement at 64 rpm: (a) simulation of fluid motion in a shake flask; (b) location of center of mass on xcoordinate (centimeters) over time (seconds), indicating coherent movement of fluid; and (c) photograph of flask in motion.
disintegrated mass from a larger-size fraction goes directly to a particular smaller-size fraction. The following are non-steady-state mass balance equations for the size fractions disintegrating in a system with a constant volume: d½w8mm ~{k1 Re½w8mm dt d½(4{8)mm ~(f 1 )k1 Re½(w8)mm{k2 Re½(4{8)mm dt d½(2{4)mm ~(f 2 )k1 Re½(w8)mmz dt (f 5 )k2 Re½(4{8)mm{k3 Re½(2{4)mm May 2012
Figure 3—Fluid movement at 200 rpm: (a) simulation of fluid motion in a shake flask; (b) location of center of mass on xcoordinate (centimeters) over time (seconds), where the circle indicates noncoherent fluid motion; and (c) photograph of flask in motion.
ð3Þ
d½(1{2)mm ~(f 3 )k1 Re½(w8)mmz(f 6 )k2 Re½(4{8)mmz dt ð6Þ (f 8 )k3 Re½(2{4)mm{k4 Re½(1{2)mm
ð4Þ
d½(v1)mm ~(f 4 )k1 Re½(w8)mmz(f 7 )k2 Re½(4{8)mmz dt
ð7Þ
(f 9 )k3 Re½(2{4)mmzk4 Re½(1{2)mm Where ð5Þ
[…] 5 the concentration of mass in the specified size fraction (g/L); 427
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Table 3—The range values of specific kinetic energy and the corresponding values of estimated characteristic flow velocity and Reynold’s number values in shake flask experiments. Rotation per minute (minute21) 64 75 100 150 200 250 300
Range of specific kinetic energy (cm/sec)2
Estimated characteristic flow velocity (cm/sec)
Estimated Reynold’s number (dimensionless)
18 117 282 317 595 1330 1400
4.2 10.8 16.8 17.8 24.4 36.5 37.5
8500 21 900 34 000 36 100 49 500 73 900 75 900
ki 5 the disintegration rate coefficient (h21) for the specified size, 1 for $8 mm, 2 for 4 to 8 mm, 3 for 2 to 4 mm, and 4 for 1 to 2 mm; and fj 5 the distribution ratio of a larger solid breaking down into smaller-solid-size fractions, with sub scripts as shown in eqs 3 through 7. The fate of a .8 mm solid is captured with eq 3, in which ki and the Reynold’s number are crucial parameters. The same rate expression with addition of distribution ratios (f1, f2, f3, and f4) is used in eqs 4 through 7 to attain the production of small-size solids (i.e., 4 to 8, 2 to 4, 1 to 2, and ,1 mm). This same approach is repeated to model the fate of intermediate-size solids in eqs 4 through 7. Using experimental results from batch experiments and the strategy for estimating model parameters presented in Karadagli et al. (2009), disintegration rate coefficients (ki) for the specified size ranges and fractional distribution ratios (fj) were estimated to determine if the disintegration model defined by eqs 3 through 7 captured all experimental trends and if turbulence had a clear effect on the disintegration rates of the flushable products. Results and Discussions Reynold’s Numbers. The Reynold’s number was used to represent turbulence in the mathematical model of disintegration. Turbulence in the experimental system was created by rotational movement of the shaker table, which moved the Fernbach flask and its contents differentially. Computational fluid dynamics modeling was used to simulate the ‘‘shake flask’’ movement at different rotational speeds and to estimate the parameters needed to define the Reynold’s number. Table 2 presents computed quasi-steady-state values of specific kinetic energy in the experimental system for rotational speeds between 64 and 300 rpm. Table 2 also provides the range of specific kinetic energy values during quasi-steady state, that is, kinetic energies fluctuated around an average value for each modeled condition and the difference between maximum and minimum specific kinetic energy values was reported as the range of specific kinetic energy. The far right column in Table 2 presents the mean specific kinetic energy of the system at quasi-steady state. The mean specific kinetic energy values in Table 2 indicate that kinetic energy in the system increased almost linearly between 64 to 150 rpm; however, a plateau is observed between 250 and 300 rpm. This plateau was caused by the transition of flow movement in the flask from a coherent pattern at low 428
rotational speeds to a chaotic and disorganized pattern at high rotational speeds. To illustrate this point, the movement of fluid within the shake flask at rotational operating speeds of 64 rpm and 200 rpm is shown graphically in Figures 2 and 3. These two rotational speeds correspond to movement mechanisms that force the fluid to move back and forth at frequencies below and above its natural frequency, respectively. When the flask was moved at rotational operating speeds less than 100 rpm, flow swirled around the bottom of the container so that its center of mass oscillated in a uniform way (Figure 2). When the flask was moved at rotational speeds greater than 100 rpm, flow moved around the base of the container in a more erratic way (Figure 3). Furthermore, the center of mass of the fluid body exhibited movement at different frequencies, and the resulting flow pattern was less orderly. As shown in frames (a) and (c) of both figures, the calculated fluid motion was similar to the observed fluid motion in the flasks. Based on these results, it was concluded that the range of specific kinetic energies is the best predictor of product disintegration (i.e., product disintegration rates, characteristic flow velocities, and Reynold’s number increased linearly with the range of specific kinetic energies). Therefore, ranges of specific kinetic energies were used to estimate characteristic flow velocities according to the second equation in Table 1. The characteristic flow velocities in the third equation in Table 1 were then used to estimate Reynold’s numbers.
Figure 4—Estimated Reynold’s number values (dimensionless) for the shake flask containing 1 L water at rotational speeds ranging from 64 to 300 rpm (minute21). Water Environment Research, Volume 84, Number 5
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Figure 5—Model simulations (lines) and experimental data (symbols) for physical disintegration of cardboard tampon applicators at 100 rpm and estimated Reynold’s number (Re) = 25 900. Concentrations are in units of grams per liter and time is in hours; (a) .8 mm (squares) and ,1 mm solids (triangles), (b) 4 to 8 mm (diamonds), and 2 to 4 mm solids (squares), and 1 to 2 mm (circles). Table 3 summarizes estimated Reynold’s number values for corresponding rotations per minute. The constant parameters for computation of Reynold’s number with eq 2 include 0.998 g/cm3 for water density or (r), 0.01 g/cm?s for dynamic viscosity of water or (m) at 20 uC, and 20.3 cm for the effective diameter of the flask (liters). The best-fit correlation between estimated Reynold’s number and rotations per minute values is illustrated in Figure 4 [i.e., Re 5 270.3 *(rpm) 2 1135.2, with R2 5 0.94]. Based on the
best-fit line, the estimated Reynold’s numbers for the experiments at 100, 150, and 200 rpm are 25 900, 39 400, and 52 900, respectively. Disintegration Results. The experimental results and model simulations obtained at 100 rpm with the cardboard tampon applicator are presented in Figure 5. The experimental results are grouped in the following size categories in millimeters: .8, 4 to 8, 2 to 4, 1 to 2, and , 1. Table 4 shows the best-fit disintegration rate constants and fractional distribution ratios for the 100-rpm experiment, for which the estimated Reynold’s number is 25 900. Figure 5 illustrates that the model simulations and experimental data match well in every size range except for one datum of 1 to 2 mm at 48 hours. This high experimental datum may have been caused by use of raw wastewater, which contains colloids that may flocculate and form solids in the 1- to 2-mm size range by 48 hours. Table 4 shows that the ki*Re values declined significantly and steadily with decreasing size. The rate for the smallest size was about 360-fold smaller than for the largest size. Because the mechanical properties of the solids from the cardboard applicator are the same, the most likely explanation for the decrease in rate constant is the different effect of turbulent forces according to chip sizes. The larger-size solids cross flow lines and are exposed to more turbulent force; as a result, they break up faster, as indicated with the rate constants. The model simulations and experimental results for 150 rpm, which had an estimated Reynold’s number of 39 400, are presented in Figures 6 and 7. Triplicate experiments were conducted at 150 rpm; hence, each data point is presented with error bars. The experimental errors increased as the amount of solid decreased in a solid category. For example, experimental errors were around 1 to 3% when large quantities of solids were measured for any size category (e.g., most of the main products remained at the early hours of the experiments). However, the experimental measurements scattered significantly when the amount of solids was very little in a size category. For instance, the amounts of the main product remaining after 2 hours into the experiment were 0 g, 0.007 g, and 0.0123 g for the triplicate experiments. These values were relatively small (,1%) when compared with the initial amount of the main product (i.e., 1.42 g), indicating that the main product disintegrated within 2 hours at 150 rpm. However, when the three measurements are compared with each other statistically, their average value, 0.0064 g, gives an error range of +/20.006 g, or 95% for these triplicate values. Therefore, the experimental data illustrated in Figures 6 and 7 indicate this variability inherent to the measurement technique. Figures 6a and 6b show, respectively, .8-mm and ,1-mm solid sizes using different time scales to highlight critical time periods
Table 4—The percent distribution of chips and disintegration rate coefficients for a cardboard tampon applicator obtained by bestfitting model simulations to the batch disintegration results at 100 rpm and estimated Reynold’s number (Re) = 25 900. Percent distribution to 21
Solid size
ki (h )
.8 mm 4 to 8 mm 2 to 4 mm 1 to 2 mm
1.8*1025 2*1026 2*1027 5*1028
May 2012
21
ki*Re (h )
4 to 8 mm
2 to 4 mm
1 to 2 mm
,1mm
0.47 0.05 5.2*1023 1.3*1023
28 -
5 20 -
2 40 20 -
65 40 80 100
429
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Figure 6—Model simulations (lines) and experimental data (symbols) for physical disintegration of cardboard tampon applicators at 150 rpm at estimated Reynold’s number (Re) = 39 400. Concentrations are in units of grams per liter and time is in hours; (a) .8 mm (squares) and , 1-mm solids (circles) and (b) the first 5 hours of data are shown in panel (a). The error bars for each datum reflect the experimental error for the triplicate measurements.
Figure 7—Model simulations (lines) and experimental data (symbols) for physical disintegration of cardboard tampon applicators at 150 rpm at estimated Reynold’s number (Re) = 39 400. Concentrations are in units of grams per liter and time is in hours; (a) 4 to 8 mm (circles), 2- to 4-mm solids (squares), and 1- to 2-mm solids (triangles); (b) the first 5 hours of data are shown in panel (a). The error bars for each datum reflect the experimental error for the triplicate measurements.
(e.g., the first 4 to 5 hours of the experiments). Figure 7 presents the data for intermediate solid sizes, which are also plotted with separate time scales (Figure 7a and 7b). Table 5 shows the best-fit disintegration rate constants and distribution ratios used for our model simulations. The trends for all sizes are similar for the three runs. The batch disintegration model captured all significant trends in the 150-rpm experiments (e.g., the rapid disintegration of the
8-mm solids in ,1 hour) (Figures 6a and 6b) and the commensurate rapid buildup of 4- to 8-mm solids in the same timeframe (Figures 7a and 7b). Similarly, the model simulated how intermediate-size solids were produced and then disintegrated by about 5 hours (Figure 7b), while the smallest size (1 to 2 mm) reached a plateau around the same time point (Figure 7a).
Table 5—The percent distribution of chips and disintegration rate coefficients for a cardboard tampon applicator obtained by bestfitting model simulations to the batch disintegration results at 150 rpm and estimated Reynold’s number (Re) = 39 400. Percent distribution to Solid size .8 mm 4 to 8 mm 2 to 4 mm 1 to 2 mm
430
ki (h21) 25
3.2*10 3*1025 8*1026 2*1027
ki*Re (h21)
4 to 8 mm
2 to 4 mm
1 to 2 mm
,1 mm
1.26 1.18 0.32 7.9*1023
37 -
5 35 -
3 5 10 -
55 60 90 100
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Figure 8—Model simulations (lines) and experimental data (symbols) for physical disintegration of cardboard tampon applicators at 200 rpm and estimated Reynold’s number (Re) = 52 900. Concentrations are in units of grams per liter and time is in hours; (a) .8 mm (squares) and ,1 mm solids (triangles); (b) the first 5 hours of data are shown in panel (a). The disintegration-rate constants (ki*Re) in Table 5 are greater than the values in Table 4, indicating that the disintegration rate increased with turbulence. The ki*Re constants in Table 5 decrease with solid size, again showing that turbulent forces have more effect on larger solids than smaller ones. The ratio of ki*Re values between the largest and smallest sizes is about 160 at 150 rpm. The experimental results and model simulations for 200 rpm are presented in Figures 8 and 9. The estimated Reynold’s number is 52 900. Similar to 150 rpm, some of the results are
Figure 9—Model simulations (lines) and experimental data (symbols) for physical disintegration of cardboard tampon applicators at 200 rpm and estimated Reynold’s number (Re) = 52 900. Concentrations are in units of grams per liter and time is in hours; (a) 4 to 8 mm (circles), 2 to 4 mm solids (diamonds), and 1- to 2-mm solids (stars); (b) the first 5 hours of data are shown in panel (a). presented with two time scales. In parallel to the other experiments, the model simulations show good agreement with experimental data for 200 rpm. Table 6 shows rate constants and distribution ratios obtained from the best fit of the experimental data. Similar to the experiments with 100 and 150 rpm, the model captured all significant trends for the 200-rpm experiments,
Table 6—The percent distribution of chips and disintegration rate coefficients for a cardboard tampon applicator obtained by bestfitting model simulations to the batch disintegration results at 200 rpm and estimated Reynold’s number (Re) = 52 900. Percent distribution to 21
Solid size
ki (h )
.8 mm 4 to 8 mm 2 to 4 mm 1 to 2 mm
7*1025 5*1025 1.5*1025 2*1027
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21
ki*Re (h )
4 to 8 mm
2 to 4 mm
1 to 2 mm
,1 mm
3.70 2.65 0.79 0.01
40 -
10 50 -
5 15 35 -
45 35 65 100
431
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Figure 10—Disintegration rate constants (ki) (1/h) vs Reynold’s number (Re) values (dimensionless): (a) .8 mm (diamonds), 8 to 4 mm (triangles), and 4 to 2 mm (squares); and (b) 2 to 1 mm (circles) solids (ki) vs estimated Reynold’s number. although the experimental datum for the ,1-mm size showed deviation for the point at 6 hours (Figure 8a). A key finding is that the time to disintegrate most of the .8-mm solids was reduced to ,0.5 hours, with the 4- to 8-mm solids peaking during the same timeframe. The ki*Re values for all the solids sizes increased noticeably from those at 100 rpm and 150 rpm; the ratio of ki*Re for the largest and smallest sizes was about 370 for 200 rpm. The percent-distribution ratios (Tables 4 through 6) of small solids during disintegration of the main products are as follows: 28 to 40% as 4- to 8-mm solids, 2 to 10% as 2- to 4-mm and 1- to 2-mm solids, and 40 to 65% as ,1-mm solids. This trend
indicates that the cardboard tampon applicator mainly disintegrates into two size categories: relatively larger solids (e.g., 4 to 8 mm) and ,10-mm solids. The same trend repeats when 4- to 8-mm solids disintegrate and generate solids of mainly the next smaller size, that is, 2 to 4 mm at 20 to 50% and ,1-mm solids at 40 to 90%. The only exception is observed during disintegration of 4- to 8-mm solids at 100 rpm, which generates 20% 2- to 4mm solids, 40% 1- to 2-mm solids, and 40% ,1-mm solids. The overall consistency in distribution ratios is expected because the solid was the same in each experiment. Hence, the effect of greater turbulence is reflected by an increasing disintegrationrate constant, not by varying distribution ratios. The disintegration-rate coefficients (ki) and (ki*Re) values for each solid size category are summarized in Table 7 for each experimental rotational speed and its corresponding Reynold’s number. The relationships between turbulence and disintegration-rate coefficients (ki) are reflected in Table 7 and illustrated in Figure 10 as ki vs estimated Reynold’s number. The ki for each solids size category increases with Reynold’s numbers, and the slopes are 1.92*1029, 1.77*1029, 5.47*10210, 5.55*10212 for solid sizes of .8, 4 to 8, 2 to 4, and 1 to 2 mm, respectively. These values indicate that the effect of turbulence is the greatest for large and intermediate sizes. Thus, once the main product breaks up, the turbulent forces can rapidly break up the intermediate size solids, as indicated by their large slope values. On the other hand, the effect of turbulence is relatively small for the smallest particles (i.e., 1- to 2-mm solids) because they can move well with water flow lines and avoid shear forces. The experimental results bear out the effect of turbulence on disintegration rates. The .8-mm solids completely disappeared in about 0.5 hours when the rotating speed was 200 rpm (Figures 8a and 8b), but the time required for 100 rpm was around 10 hours (Figure 5a). This difference reflects the 8-fold higher value of ki*Re for the .8-mm solids when comparing 200 rpm to 100 rpm (Tables 4 and 6). The rapid rise and fall in concentrations of the 4- to 8-mm and 2- to 4-mm sizes in Figures 6 and 8 confirms that they also disintegrate relatively rapidly and are affected strongly by turbulence. The smallest size (i.e., 1- to 2-mm solids) attained a plateau concentration in all experiments because its disintegration rate is much slower than that of the larger sizes. Conclusions The physical disintegration model captured all the main features of the experimental results for the disintegration kinetics of cardboard tampon applicators when the rotating speed was increased from 100 to 200 rpm, which corresponds to an increase of estimated Reynold’s number from 25 900 to 52 900. The disintegration rates of solids depended strongly on turbulence, as
Table 7—Values for the disintegration rate coefficients (ki) and (ki*Re) for the solid size ranges obtained at 100, 150, and 200 rpm corresponding to estimated Reynold’s number (Re) values of 25 900, 34 900, and 52 900, respectively. 100 rpm (Re = 25 900) Solid size (mm) .8 4 to 8 2 to 4 1 to 2
432
21
ki (h ) 1.8*1025 2*1026 2*1027 5*1028
150 rpm (Re = 39 400) 21
ki (h )
0.47 0.05 5.2*1023 1.3*1023
3.2*1025 3*1025 8*1026 2*1027
ki*Re (h )
21
200 rpm (Re = 52 900) 21
ki (h21)
1.26 1.18 0.32 7.9*1023
7*1025 5*1025 1.5*1025 2*1027
ki*Re (h )
ki*Re (h21) 3.70 2.65 0.79 0.01
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represented by estimated Reynold’s number, although the distribution ratios remained similar. These results indicate that the disintegration model gives a good representation of solid disintegration from larger to smaller chips. The new results demonstrate that increasing turbulence increased the estimated disintegration rate constants, which shortened the disintegration time. Acknowledgments The authors would like to thank the Procter & Gamble Company for their support in this research and to Erin Schwab for her part in conducting the laboratory disintegration experiments. Submitted for publication January 9, 2011; revised manuscript submitted December 14, 2011; accepted for publication February 22, 2012. References Anderson, J. A. (1995) Computational Fluid Dynamics: The Basics with Applications, 6th ed.; McGraw-Hill: New York. Hirt, C. W.; Nichols, B. D. (1981) Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Computational Physics, 39, 201–225.
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Hirt, C. W.; Sicilian, J. M. (1985) A Porosity Technique for the Definition of Obstacles in Rectangular Cell Meshes. Proceedings of the 4th International Conference on Numerical Ship Hydrodynamics; Washington, D.C., Sept 24–27; National Academy of Sciences: Washington, D.C. Karadagli, F.; Rittmann, B. E.; McAvoy, D. (2009) Development of Mathematical Model for Physical Disintegration of Flushable Consumer Products in Sewerage Systems. Water Environ. Res., 81 (5), 459–465. Rittmann, B. E.; McCarty, P. L. (2001) Environmental Biotechnology: Principles and Applications; McGraw-Hill: New York. Stumm, W; Morgan, J. J. (1996) Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, 3rd ed.; Wiley & Sons: New York. Sicilian, J. M.; Hirt, C. W.; Harper, R. P. (1987) FLOW-3D: Computational Modeling Power for Scientists and Engineers; Flow Science Report (FSI–87–00–1); Flow Science, Inc.: Santa Fe, New Mexico. Yakhot, V.; Orszag, S. A. (1986) Renormalization Group Analysis of Turbulence—I. Basic Theory. J. Sci. Computing, 1 (1), 3–51. Yakhot, V.; Smith, L. M. (1992) The Renormalization Group, the eExpansion and Derivation of Turbulence Models. J. Sci. Computing, 7 (1), 35–61.
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